Abstract
In this paper, a nonseparation approach is developed to explore the problem of finite-time and fixed-time synchronization for fully complex-valued dynamical networks. Firstly, to avoid the traditional separation method, the signum function of complex-valued vectors is proposed as an extension of the real-valued signum function and several essential properties of it are derived. Based on the introduced signum function, several complex-valued controllers are directly designed on the networks, which are totally different from the existing real-valued control designs on the decomposed subsystems. Furthermore, by directly constructing Lyapunov functions in the complex field and developing some innovative inequalities, several criteria described by matrix inequalities are derived to achieve finite-time or fixed-time synchronization, which are much simpler and more convenient compared with the previous algebraic conditions. Finally, some numerical simulations are provided to support the validity of the derived results.
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